They won’t now complicate a Hillary Clinton presidency, but the emails from her time in the State Department will still trickle out over the next months as the administration tries to clean up the mess Mrs. Clinton left.

Though President-elect Donald Trump has signaled that he would halt any official federal investigations into Mrs. Clinton’s activities as secretary of state, groups that have driven the issue say they will continue to press the legal cases that have forced the release of tens of thousands of her emails and could still pry loose thousands more.

“Nothing has been settled,” said Tom Fitton, president of Judicial Watch, one of the groups that has pressed for more transparency from the Obama administration. He said the issues go much deeper than Mrs. Clinton alone.

The fight for transparency in Mrs. Clinton’s emails has broken major ground in open-records laws, with several rounds of legal battles to come.

Both sides will be in U.S. District Court for the District of Columbia on Tuesday to argue over some emails that the State Department withheld. Judicial Watch says the emails show evidence of misconduct at the department.

On tap for later are arguments over whether the government must release the messages Mrs. Clinton exchanged with President Obama from her secret account and fights over the tens of thousands of messages the FBI recovered during its criminal probe into the emails.

Congress also is likely to do some more investigating, though how much time it spends remains to be seen.

Rep. Jason Chaffetz, Utah Republican and chairman of the House Oversight and Government Reform Committee, said after the elections that his investigation will proceed because the public still deserves to see the whole picture of wrongdoing.

“It was never about the political targeting of Hillary Clinton. But she did create one the biggest security breaches in the history of the State Department,” he said on Fox News.

During his campaign against Mrs. Clinton, Mr. Trump said he would put her behind bars, and crowds at his rallies chanted, “Lock her up.” Now, the president-elect says he has lost interest.

“I don’t want to hurt the Clintons, I really don’t,” Mr. Trump told The New York Times. “She went through a lot and suffered greatly in many different ways, and I am not looking to hurt them at all. The campaign was vicious.”

Mr. Fitton said it was a mistake for the FBI to clear Mrs. Clinton and that the investigations are not about political retribution but rather about holding officials accountable.

“She was the head of an agency, and she had the help of agency officials in avoiding the rule of law here,” he said. “There was a massive fraud committed on the American people by federal officials, some of whom are going to still be there in a Trump administration. Who in the State Department is responsible for this cover-up? Who in the Justice Department is responsible for this?”

Judicial Watch is trying to force John Bentel, the man who was in charge of preserving records for Mrs. Clinton in the State Department, to answer questions. Mr. Bentel cited his Fifth Amendment right against self-incrimination and refused to talk, but Judicial Watch says he can’t face legal jeopardy and should be forced to testify because there is no active criminal investigation.

Mr. Fitton’s organization is also trying to make Mrs. Clinton expand on written testimony that she filed in one of the cases.

Under questioning earlier this year, the former secretary admitted that nobody gave her permission to use a secret server — contradicting her public assertions.

But Mrs. Clinton refused to answer several questions about whether she had been aware of security warnings discouraging the use of private email accounts, nor would she answer how she came to testify to Congress that up to 95 percent of her messages were already stored in other State Department accounts. She cited attorney-client privilege in refusing to answer that latter question.

In a filing last week, Mrs. Clinton’s attorneys said questions about whether Mrs. Clinton knew of the dangers of using her own BlackBerry device were irrelevant to the open-records lawsuit Judicial Watch filed.

The attorneys also indicated that they were the ones who informed Mrs. Clinton of the 95 percent number — a figure they tallied by looking at the recipients of her messages.

“Secretary Clinton answered the interrogatories to the best of her knowledge in good faith, and her answers are consistent with the voluminous public record,” the attorneys said.

The Obama administration has defended Mrs. Clinton’s position throughout the proceedings.

Outside groups are calling on the incoming Trump administration to change that approach and to accede to more transparency rather than fighting it.